The descriptive complexity of the set of arc-connected compact subsets of the plane

TL;DR

This paper precisely determines the descriptive set-theoretic complexity of the set of all arc-connected compact subsets of the plane, revealing it exceeds classical analytic and coanalytic classes but is below the class for three-dimensional cases.

Contribution

It computes the exact descriptive complexity of arc-connected compact subsets of the plane, establishing a new position in the hierarchy of set-theoretic complexity.

Findings

The set's complexity is higher than $oldmath oldsymbol ext{ extSigma}^1_1$ and $oldmath oldsymbol extPi^1_1$.

The set's complexity is lower than $oldmath oldsymbol extPi^1_2$.

It differentiates the complexity of planar and spatial arc-connected sets.

Abstract

We compute the exact complexity of the set of all arc-connected compact subsets of $\boldmath R^2$, which turns out to be strictly higher than the classical $\boldmath \Sigma^1_1$ and $\boldmath \Pi^1_1$ classes of analytic and coanalytic sets, but stricly lower than the class $\boldmath \Pi^1_2$ which is the exact descriptive class of the set of all arc-connected compact subsets of $\boldmath R^3$.